Δ4

Δ2

Δ3

Ruudu ABCD tipud on vastupäeva ühendatud vastaskülgede
 keskpunktidega nii, nagu on näidatud  kõrvaloleval joonisel.
 Tulemusena on saadud väiksem ruut EFGH. Tõestame, et
 ruutude EFGH ja ABCD pindalade suhe on 1:5.

Δ1

\\\frac{S_{EFGH}}{S_{ABCD}}=\frac{1}{ 5}

Animeeri tõestus, libistades liugurit 

p: y = 4
B = Point(p)
A = Point(p)
hulknurk1, f, g, h, i, C, D = Polygon(A, B, 4)
J = Midpoint(h)
K = Midpoint(g)
I = Midpoint(i)
j = Segment(B, J)
k = Segment(C, I)
m = Segment(A, K)
F = Intersect(m, j)
G = Intersect(k, j)
l = Segment(A, B)
L = Midpoint(l)
n_{1} = Segment(D, L)
H = Intersect(n_{1}, k)
E = Intersect(n_{1}, m)
q1, h_1, e, f_1, g_1 = Polygon(H, E, F, G)
n = 4 (slider from 1 to 4, step 1)
α = 3.141592653589793 rad (slider from 0° to 180°, step 0.017453292519943295)
W = Rotate(E, α, L)
p_{1}: y = 4
q = Segment(L, W)
p_{2}: y = 4
p_{3}: y = 4
p_{4}: y = 4
B' = Rotate(B, α, K)
F' = Rotate(F, α, K)
a = Segment(K, F')
c = Segment(F', B')
d = Segment(B', K)
G' = Rotate(G, α, J)
C' = Rotate(C, α, J)
j_1 = Segment(J, G')
k_1 = Segment(G', C')
q_1 = Segment(C', J)
D' = Rotate(D, α, I)
H' = Rotate(H, α, I)
r_1 = Segment(I, H')
s_1 = Segment(H', D')
t_1 = Segment(D', I)
q3, h_3, h', d', e_2 = Polygon(H, H', D', E)
q4, a_1, b_1, c_1, d_1 = Polygon(B', G, F, F')
q5, g', c', h_4, g_3 = Polygon(G', C', H, G)
i_2 = Segment(C', J)
A' = Rotate(A, α, L)
r = Segment(L, A')
M = Point(q)
s = Segment(M, A')
o: hulknurk1 / q1
q2, e_1, m_1, a', f_2 = Polygon(E, M, A', F)
b: q2 + q3 + q4 + q5
t: b + q1
N = (21.70831932773109, 2.675614824054623)
O = (22.218823529411758, 2.675614824054623)
hulknurk2, i_1, l_1, g_2, h_2, P, Q = Polygon(N, O, 4)
tekst2: "1. \;   \; S_{EFGH}=" + (LaTeX[q1]) + " \\\\"
tekst3: "2. \;  \; S_{Δ1}+S_{Δ2}+S_{Δ3}+S_{Δ4}=\\ \;  \; =" + (LaTeX[q2]) + "+" + (LaTeX[q3]) + "+" + (LaTeX[q4]) + "+" + (LaTeX[q5]) + "=" + (LaTeX[b]) + "\\\\3. \;  \; " + (LaTeX[q1]) + "+" + (LaTeX[b]) + "=" + (LaTeX[t]) + "=S_{ABCD}"